Contribution of time delays to p53 oscillation in DNA damage response

Abstract

Although the oscillatory dynamics of the p53 network have been extensively studied, the understanding of the mechanism of delay‐induced oscillations is still limited. In this paper, a comprehensive mathematical model of p53 network is studied, which contains two delayed negative feedback loops. By studying the model with and without explicit delays, the results indicate that the time delay of Mdm2 protein synthesis can well control the pulse shape but cannot induce p53 oscillation alone, while the time delay required for Wip1 protein synthesis induces a Hopf bifurcation to drive p53 oscillation. In addition, the synergy of the two delays will cause the p53 network to oscillate in advance, indicating that p53 begins the repair process earlier in the damaged cell. Furthermore, the stability and bifurcation of the model are addressed, which may highlight the role of time delay in p53 oscillations.

1 Introduction

The tumour suppressor p53 is the centre of the cell signalling network and can be activated by DNA damage caused by $\gamma$ radiation [1]. p53 is a key tumour suppressor, whose gene mutation is present in more than half of human cancers, resulting in a reduction in or loss of its transcriptional function [2]. Under normal conditions, p53 keeps a low level because of the inhibitory effects provided by its negative regulators [3]. In response to acute stress, p53 as a transcription factor can regulate the expression of downstream target genes to induce cell survival or death [4]. It suggested that how to choose the outcomes is primarily governed by p53 levels and pulse numbers [5, 6]. On the one hand, low levels of p53 can induce cell‐cycle arrest and promote DNA repair to lead to cell survival, and high levels of p53 can induce apoptosis to destroy irreparably damaged cells. On the other hand, transient p53 pulses induce cell life and sustained p53 pulses lead to cell death, which avoids premature apoptosis resulting from unexpectedly high levels of p53 [7]. Owing to the pivotal role of the p53 levels and p53 pulses in mediating cell fate decisions, many efforts have been made in industry and academia to develop new and effective p53‐based anticancer therapies [8].

It is widely suggested that negative feedback loop can induce biochemical oscillation [9, 10, 11, 12], and the functional mechanism of such feedback loop in the p53 network has been studied by many researchers [6, 13, 14, 15]. It was reported that p53 dynamics depend on the p53‐murine double minute 2 (Mdm2) and ataxia telangiectasia mutated (ATM)‐wild‐type p53‐induced phosphatase 1 (ATM‐p53‐Wip1) negative feedback loops [16]. The p53‐Mdm2 negative feedback loop is the base of the generation of p53 oscillations [17, 18]. Mdm2 protein is the centre of the p53 regulatory network and can inhibit p53 activity in two ways [19]. First, Mdm2 can directly interfere with the ability of p53 to induce gene expression [20], and second, Mdm2 can eliminate p53 through proteolytic degradation [21]. In particular, Mdm2 protein can be phosphorylated by ATM to form two distinct forms [22]. Besides, the ATM‐p53‐Wip1 negative feedback loop can induce uniform p53 pulses [23]. Under acute stress, active p53 induces transcription and translation of Wip1 gene to form Wip1 protein [24], in turn, Wip1 protein can reduce ATM activation [25]. Therefore, it is necessary to further study the oscillation mechanism of the negative feedback loop in the p53 network to better understand how the p53 dynamics are regulated.

Time delays have proven to be a key factor in modelling, designing, and controlling regulatory networks [26]. There are two different timescales in the p53 network [27], the protein–protein binding or unbinding reactions are fast and the transcription and translation reactions are slow. In general, the time delay required for the protein–protein reaction is always omitted because it is much smaller than the time delay required for protein synthesis. Since the transcriptional and translational processes are slow and complex, the time delays in gene expression are inevitable [28]. Typically, there is a delay of about 0.7 h between the initiation of transcription of the Mdm2 gene and the appearance of mature Mdm2 protein. Similarly, there is a delay of about 1.25 h from Wip1 gene to the intact Wip1 protein [29]. Previously, some models have explored the oscillation mechanism of the p53 network [6, 29, 30, 31]. Such as LanMa found the number of pulses increases with ionising radiation (IR) dose [6], Wagner demonstrated how to effectively dampen the negative feedback loop to produce limit‐cycle oscillations [30], and Purvis studied how to change p53 levels to control the cell fate in response to DNA damage [29]. However, most studies are based on experiments and simulations and focus on the effects of model parameters. For simplicity, these models implicitly assume that all fast and slow reactions in the p53 network are instantaneous. The precise effects of each time delays are still not well‐understood; especially the Hopf bifurcation caused by time delay is ignored. Meanwhile, some models ignore the effects of ATM‐p53‐Wip1 negative feedback loop. Therefore, further research is expected to more quantitatively and comprehensively understand the effect of time delay on p53 oscillations, especially when multiple delays are present.

Motivated by the above considerations, we develop a more accurate delayed mathematical model to investigate how the time delays required for Mdm2 and Wip1 protein synthesis mediate the p53 dynamics when cells respond to $\gamma$ radiation. In particular, our model incorporates five main components: p53 protein; two forms of Mdm2, namely unmodified Mdm2 ($\text{Mdm}{2}_r$) and modified Mdm2 ($\text{Mdm}{2}_m$); Wip1 protein; and ATM protein. Our results indicate that time delays are essential for p53 oscillations, especially, a different type of time delays have different functions on p53 dynamics. The time delay required for Mdm2 protein synthesis is primarily used to control the shape of pulses and the time delay required for Wip1 protein synthesis is primarily used to induce oscillations. In addition, the synergy of the two time delays will cause the system to oscillate in advance compared with a single time delay, indicating that p53 begins the repair process earlier in the damaged cell. Furthermore, considering the time delay as the bifurcation parameter, the supercritical Hopf bifurcation is proved by analysing the characteristic equation of the model.

2 Model formulation

The interactions among ATM, p53, Wip1, and Mdm2 are provided in Fig. 1. DNA damage induced by sustained $\gamma$ radiation can bind to DNA repair proteins to form protein complexes which are sensed by ATM, a protein kinase that activates p53 [6]. Subsequently, active p53 induces the transcription and translation of Mdm 2 and Wip 1 genes to form Mdm2 and Wip1 proteins. Moreover then, Mdm2 protein can be rapidly modified by ATM to form modified Mdm2 ($\text{Mdm}{2}_m$). The unmodified Mdm2 ($\text{Mdm}{2}_r$) promotes a fast degradation of p53 and the modified Mdm2 has a weaker inhibition on p53 [22]. At the same time, the Wip1 protein can inhibit the activity of ATM, thereby forming a short negative feedback loop and a long negative feedback loop, namely p53‐Mdm2 and ATM‐p53‐Wip1. Importantly, protein synthesis is a complex and time‐consuming process that requires a certain time to complete transcriptional and translational regulations. Therefore, there are inevitable time delays from the initiation of transcription to the appearance of the intact protein. As shown in Fig. 1, ${\tau }_1$ and ${\tau }_2$ represent the total time required for transcription and translation of the Mdm2 protein and Wip1 protein, respectively. On the basis of these properties, we modify the model of Geva‐Zatorsky, which is used to describe the dynamics of the p53 network in response to $\gamma$ radiation, the details are as follows:

$$\left\{\begin{array}{c}\frac{\mathrm{d}p(t)}{\mathrm{d}t}={\beta }_p-{\alpha }_{r}p(t)x(t)-{\alpha }_{m}p(t)y(t),\\ \frac{\mathrm{d}x(t)}{\mathrm{d}t}={\beta }_{m}p(t-{\tau }_1)-\beta \frac{s^{n_1}(t)}{s^{n_1}(t)+T_s^{n_1}}x(t)-\alpha x(t),\\ \frac{\mathrm{d}y(t)}{\mathrm{d}t}=\beta \frac{s^{n_1}(t)}{s^{n_1}(t)+T_s^{n_1}}x(t)-\alpha y(t),\\ \frac{\mathrm{d}w(t)}{\mathrm{d}t}={\beta }_{w}p(t-{\tau }_2)-{\alpha }_{w}w(t),\\ \frac{\mathrm{d}s(t)}{\mathrm{d}t}={\beta }_s-k\frac{w^{n_2}(t)}{w^{n_2}(t)+T_w^{n_2}}s(t)-{\alpha }_{s}s(t),\end{array}\right.$$ (1)

where p, x, y, w, and s represent p53, unmodified Mdm2, modified Mdm2, Wip1, and ATM, respectively. In our model, the positive items represent promotion and the negative items represent inhibition and self‐degradation. In particular, Table 1 gives the complete list of model parameters and their default values, which are taken from [27, 29].

Fig. 1.

DNA damage induced by sustained

$\gamma$

radiation can bind to DNA repair proteins to form protein complexes which actives ATM and p53. Subsequently, active p53 induces transcription and translation of Mdm2 and Wip1 genes, forming Mdm2 and Wip1 proteins, where

${\tau }_1$

and

${\tau }_2$

represent inevitable time delays. In addition, the Mdm2 protein can be modified by ATM and promotes the degradation of p53. Meanwhile, Wip1 protein can inhibit the activity of ATM, thereby forming two negative feedback loops

Table 1.

Parameters values for the mathematical model

Parameter	Description	Value
${\beta }_p$	p53 production rate	$0.9{\mathrm{h}}^{-1}$
${\beta }_m$	$\text{Mdm}{2}_r$ production rate	$1.2{\mathrm{h}}^{-1}$
$\beta$	$\text{Mdm}{2}_m$ production rate	$140{\mathrm{h}}^{-1}$
${\beta }_w$	Wip1 production rate	$0.25{\mathrm{h}}^{-1}$
${\beta }_s$	ATM production rate	$10{\mathrm{h}}^{-1}$
${\alpha }_r$	$\text{Mdm}{2}_r$ ‐induced p53 degradation	$140{\mathrm{h}}^{-1}$
${\alpha }_m$	$\text{Mdm}{2}_m$ ‐induced p53 degradation	$1.4{\mathrm{h}}^{-1}$
$\alpha$	Mdm2 degradation rate	$1{\mathrm{h}}^{-1}$
${\alpha }_w$	Wip1 degradation rate	$0.7{\mathrm{h}}^{-1}$
${\alpha }_s$	ATM degradation rate	$7.5{\mathrm{h}}^{-1}$
k	Wip‐induced inactivation of ATM	$50{\mathrm{h}}^{-1}$
${\tau }_1$	time delay of Mdm2 protein synthesis	0.7 h
${\tau }_2$	time delay of Wip1 protein synthesis	1.25 h
$n_1$	Hill coefficient of $\text{Mdm}{2}_r$ production	4
$n_2$	Hill coefficient of ATM degradation	4
$T_s$	half‐maximal p53 activation threshold	1
$T_w$	half‐maximal ATM inhibition threshold	0.2

Noteworthy, our model is basically similar to the model of Geva‐Zatorsky but there are several key differences. Since the nature of Wip1 was unclear, Geva‐Zatorsky modelled Wip1 using the identical dynamics as Mdm2 [27]. By contrast, our model has specific parameters for Wip1 such as the total of transcriptional and translational time delay ${\tau }_2$, production rate ${\beta }_w$, and degradation rate ${\alpha }_w$. Furthermore, ${\alpha }_s$ is added to describe the self‐degradation rate of ATM, and Michaelis equation is added to describe the interactions of Wip1 and ATM [32]. Importantly, the model emphasises the role of Mdm2, which will explain the oscillation mechanism of the p53 network from another perspective.

3 Main results

In this section, we study the effects of time delays required for Mdm2 and Wip1 protein synthesis on p53 dynamics. In this paper, the parameter values used in all calculations are listed in Table 1. Accordingly, we can easily calculate the positive equilibrium point of system (1) is $E^{\ast }$  = (0.3497, 0.0143, 0.4053, 0.1249, 0.7093) by using the software Mathematica 10. In addition, we always use Mdm2 to represent the sum of unmodified Mdm2 and modified Mdm2, unless otherwise stated.

3.1 Dynamics of the p53 network without time delay

To determine the importance of such time delays in the p53 network, it is important to study the dynamical behaviour of the p53 network without time delay. Therefore, we first study the dynamics of a p53 network with ${\tau }_1={\tau }_2=0$. Fig. 2 shows the outcomes of the p53 network without time delay. Fig. 2 a shows that the level of p53 rapidly rises and reaches a peak 0.58 at 1.18 h, and then its level quickly decreases to the stable level. At the same time, Mdm2 rises and reaches a peak 0.49 at 1.94 h, then quickly falls to the stable level. Importantly, Fig. 2 a also shows that p53 and Mdm2 only undergo one pulse, and finally the stable level of Mdm2 is higher than that of p53, which is consistent with the calculation. Furthermore, Fig. 2 b shows that no limit‐cycle oscillations are produced, indicating that the system is asymptotically stable when ${\tau }_1={\tau }_2=0$. In summary, we conclude that the p53 network is asymptotically stable without time delay, which is consistent with the experimental conclusions [33].

Fig. 2.

P53 network is asymptotically stable when

${\tau }_1={\tau }_2=0$

(a) The concentrations of p53 and Mdm2 rapidly stabilised, (b) p53 network is stable when there is no time delay

3.2 Effects of time delay required for Mdm2 protein synthesis

It has been reported that we can ignore the delays required for transcription and translation only when other timescales are much larger than it [34]. However, when the system has multiple delays of the same timescale, we are still unclear about the specific effects of each delay. For this purpose, we next separately study the effects of ${\tau }_1$ and ${\tau }_2$. Mdm2 is the most important regulator of p53 and the core of the p53 regulatory network. Studies have shown that p53‐Mdm2 negative feedback is the basis for p53 oscillations [18]. To further investigate the specific effects of the time delay required for Mdm2 protein synthesis in this complex network, Fig. 3 shows the effects of ${\tau }_1$ on the dynamical behaviour of p53 network.

Fig. 3.

Effects of

${\tau }_1$

on the dynamical behaviour of p53 network

(a–c) When ${\tau }_2=0$, ${\tau }_1$ can change the peaks of pulses but cannot change the stability of the p53 network, (d–f) When ${\tau }_2=1.25$, ${\tau }_1$ can induce oscillations and control the amplitude and period of oscillations

In the first place, Figs. 3 a–c show the effects of different ${\tau }_1$ on the p53 network when ${\tau }_2=0$. It is easy to see that it is asymptotically stable regardless of how ${\tau }_1$ changes and the peak is increased with ${\tau }_1$. In particular, Fig. 3 b shows a distinct difference from the others, the Mdm2 level rises after a different delay, which is caused by the time delay required for Mdm2 protein synthesis. In the second place, we let ${\tau }_2=1.25$, the intrinsic time delay in Wip1 protein synthesis. Figs. 3 d–f show that when ${\tau }_1=0$ the p53 network is stable, whereas oscillatory when ${\tau }_1=0.5$ and ${\tau }_1=1$. In addition, there are 36 pulses when ${\tau }_1=0.5$ and 35 pulses when ${\tau }_1=1$. Moreover, the amplitude and period of oscillations are increased with ${\tau }_1$. To sum up, we can conclude that the time delay required for Mdm2 protein synthesis cannot induce oscillations alone but can control the amplitude and period of oscillations. This conclusion is further refined from the previous work, that is, not all time delays can induce oscillations alone.

3.3 Effects of time delay required for Wip1 protein synthesis

It reported that ATM‐p53‐Wip1 negative feedback loop can induce uniform p53 pulses [23]. However, the effects of time delay required for Wip1 protein synthesis on the generation of p53 oscillations have not been studied. As a supplement to existing researches, we next study the effects of ${\tau }_2$ in the generation of oscillations.

First of all, Figs. 4 a–c show the effects of different ${\tau }_2$ on the p53 network when ${\tau }_1=0$. Fig. 4 a indicates that ${\tau }_2$ cannot change the peak of the first pulse of p53. Fig. 4 c indicates that the Wip1 will start to rise after a delay of ${\tau }_2$. From Figs. 4a–c, we can get that p53 network is stable when ${\tau }_2=0.5$ and ${\tau }_2=1$, while oscillatory when ${\tau }_2=1.5$, indicating that the p53 network experiences a supercritical Hopf bifurcation to oscillate. Taking ${\tau }_2$ as the bifurcation parameter, we get the bifurcation point ${\tau }_{\ast }=1.43$, and the detailed calculation processes are shown in Section 5.1, which is consistent with the simulations. Similarly, letting ${\tau }_1=0.7$, the intrinsic time delay in Mdm2 protein synthesis, Figs. 4 d–f show that when ${\tau }_2=0<0.133$, the p53 network is asymptotically stable, whereas oscillatory when ${\tau }_2=0.2>0.133$, indicating that the p53 network also undergoes a Hopf bifurcation to drive p53 oscillations at ${\tau }_2=0.133$. Moreover, the amplitude and period of pulses are also increased with ${\tau }_2$. Taken together, we can conclude that the time delay required for Wip1 protein synthesis can induce p53 oscillations alone and also controls the amplitude and period.

Fig. 4.

Effects of

${\tau }_2$

on the dynamical behaviour of the p53 network

(a–c) When ${\tau }_1=0$, the system undergoes a supercritical Hopf bifurcation to the driven p53 network to be oscillatory at ${\tau }_2=1.43$, (d–f) When ${\tau }_1=0.7$, p53 network also undergoes a Hopf bifurcation to driven p53 oscillations at ${\tau }_2=0.133$

3.4 Effects of ${\tau }_1$ and ${\tau }_2$ coexistence on the p53 network

Actually, ${\tau }_1$ and ${\tau }_2$ are coexisting in p53 network. So, we also need to study the effects of the coexistences of ${\tau }_1$ and ${\tau }_2$ on the p53 network. For simplifying the calculation process and highlighting the purpose of the study, we always assume ${\tau }_1={\tau }_2=\tau$ in the following calculations and simulations. Taking $\tau$ as a bifurcation parameter, we calculate the bifurcation point ${\tau }_0=0.552$. As an example, Fig. 5 shows the numerical simulation results when $\tau =0.5\text{and}0.6$. We can see that the p53 network is asymptotically stable when $0<\tau <{\tau }_0$, whereas oscillatory when $\tau >{\tau }_0$. The results indicate that time delay is essential for oscillation, and when $\tau$ crosses the critical value ${\tau }_0$, the p53 network loses the stability and undergoes a supercritical Hopf bifurcation. In addition, we find that ${\tau }_0$ is smaller than ${\tau }_{\ast }$, which indicates that the combined effects of ${\tau }_1$ and ${\tau }_2$ would cause the p53 network to oscillate in advance, leading to the damaged cells to begin the repair process earlier.

Fig. 5.

Effects of time delays on the p53 network when ${\tau }_1={\tau }_2=\tau$. The figures show that the system is asymptotically stable when $\tau =0.5<{\tau }_0=0.552$ and oscillatory when $\tau =0.6>{\tau }_0=0.552$. The result indicates that as $\tau$ increases and crosses the critical value ${\tau }_0$, the system loses the stability and undergoes a supercritical Hopf bifurcation

(a, b) P53 network is asymptotically stable when τ  = 0.5 < τ ₀  = 0.552, (c, d) p53 network is oscillatory when τ  = 0.6 > τ ₀  = 0.552

To verify the validity of our model, Fig. 6 shows the time courses of the p53 network with time delay. Comparing the different forms of Mdm2, we find that $\text{Mdm}{2}_r$ and $\text{Mdm}{2}_m$ have the same pulse numbers and reach the peak and nadir at the same time. Specially, the level of $\text{Mdm}{2}_r$ is smaller than $\text{Mdm}{2}_m$ (Fig. 6 a). Over the first pulse, Fig. 6 b shows that the ATM levels first rise rapidly to peak 1.33, and keeps the level for a while and then falls. Similarly, p53 follows ATM to peak 0.85 and then decline simultaneously. In particular, they always keep rising and falling at the same time, and have the same pulse number. In addition, Wip1 and Mdm2 act as negative regulators, they have the same behaviour. After a time delay, they rise and peak simultaneously, and they also have the same pulse number. Importantly, over the same period, the rise and fall of ATM and p53 are the opposite of Wip1 and Mdm2, which is consistent with the actual situation.

Fig. 6.

Time courses of the p53 network with time delay

(a) Level of $Mdm{2}_r$ and $Mdm{2}_m$ with $\tau =0.6$, (b) Time courses of p53 network with $\tau =0.6$

4 Conclusion

In previous works, they claimed that different protein production rates and degradation rates in the feedback loop lead to different dynamical behaviours [35]. At the same time, though some studies have also addressed the role of time delays, they studied the time delay in a general way and argued that the effects of different time delays are the same. Different from previous works, we separately studied the dynamics of the mathematical model when ${\tau }_1$ or ${\tau }_2$ exist alone and the results were compared with the results of the simultaneous presence of ${\tau }_1$ and ${\tau }_2$. More importantly, our model emphasises the role of Mdm2 by considering the two forms of Mdm2, and the Hopf bifurcation driven by time delay is proved by analysing the relevant characteristic equations.

In this paper, a delayed mathematical model about the p53 network was developed to understand the different contributions of the time delays required for Mdm2 and Wip1 protein syntheses to the generation of p53 oscillations. At the first place, we found that the p53 network is always stable without time delay. Next, by studying the effects of ${\tau }_1$ and ${\tau }_2$ separately, the results indicate that ${\tau }_1$ can control the shape of pulses but cannot induce oscillations alone, and ${\tau }_2$ can induce a supercritical Hopf bifurcation at ${\tau }_{\ast }=1.43\mathrm{h}$ to drive the p53 network to be oscillatory. Furthermore, considering ${\tau }_1$ and ${\tau }_2$ coexist in the p53 network, we calculated the critical value of time delay ${\tau }_0=0.552$ when ${\tau }_1={\tau }_2=\tau$. As $\tau$ increases and crosses the critical value ${\tau }_0$, the p53 network loses the stability and also undergoes a supercritical Hopf bifurcation. In particular, ${\tau }_0$ is smaller than ${\tau }_{\ast }$, which indicates that the damaged cells start to repair process earlier.

For the above, the results indicate that the time delays are essential for oscillations of p53 network and can be used to control the amplitude and period of oscillations. In particular, different time delays have different functions in regulating p53 dynamics, i.e. the time delay required for Wip1 protein synthesis is essential for the generation of p53 oscillations and the time delay required for Mdm2 protein synthesis can well control the shape of pulses. On the basis of these findings, some valuable insights for cancer treatment of the p53 pathway may be provided.

5 Stability and Hopf bifurcation

In this section, we use the Hopf bifurcation theory and method [36, 37, 38] to analyse the dynamics of system (1) driven by time delays. Since ${\tau }_1$ cannot change the stability of the system alone, we omit the theoretical analysis of ${\tau }_1$.

5.1 Effects of ${\tau }_2$ on p53 network

To highlight the effects of ${\tau }_2$, we assume ${\tau }_1=0$. Obviously, the system has a unique positive equilibrium denoted as $E^{\ast }=(p^{\ast },x^{\ast },y^{\ast },w^{\ast },s^{\ast })$. Let $\overline{p}(t)=p(t)-p^{\ast }$, $\overline{x}(t)=x(t)-x^{\ast }$, $\overline{y}(t)=y(t)-y^{\ast }$, $\overline{w}(t)=w(t)-w^{\ast }$, $\overline{s}(t)=s(t)-s^{\ast }$, and still denote $\overline{p}(t)$, $\overline{x}(t)$, $\overline{y}(t)$, $\overline{w}(t)$, $\overline{s}(t)$ by p, x, y, w, s, then system (1) becomes

$$\left\{\begin{array}{c}\frac{\mathrm{d}p(t)}{\mathrm{d}t}=-({\alpha }_r{x}^{\ast }+{\alpha }_m{y}^{\ast })p(t)-{\alpha }_r{p}^{\ast }x(t)-{\alpha }_m{p}^{\ast }y(t),\\ \frac{\mathrm{d}x(t)}{\mathrm{d}t}={\beta }_{m}p(t)-(\alpha +\beta f(s^{\ast }))x(t)-\beta f^{\mathrm{\prime }}(s^{\ast })x^{\ast }s(t)\\ -\beta f^{\mathrm{\prime }}(s^{\ast })s(t)x(t)-\beta (s+s^{\ast }){\mathrm{\Sigma }}_{i=2}^{\mathrm{\infty }}\frac{1}{i!}{f}^{(i)}(s^{\ast })s^i(t)\\ \frac{\mathrm{d}y(t)}{\mathrm{d}t}=\beta f(s^{\ast })x(t)+\beta f^{\mathrm{\prime }}(s^{\ast })x^{\ast }s(t)-\alpha y(t)+\beta f^{\mathrm{\prime }}\\ (s^{\ast })x^{\ast }s(t)x(t)+\beta (s+s^{\ast }){\mathrm{\Sigma }}_{i=2}^{\mathrm{\infty }}\frac{1}{i!}{f}^{(i)}(s^{\ast })s^i(t),\\ \frac{\mathrm{d}w(t)}{\mathrm{d}t}={\beta }_{w}p(t-{\tau }_2)-{\alpha }_{w}w(t),\\ \frac{\mathrm{d}s(t)}{\mathrm{d}t}=-k{g}^{\mathrm{\prime }}(w^{\ast })s^{\ast }w(t)-(kg(w^{\ast })+{\alpha }_s)s(t)-k{g}^{\mathrm{\prime }}\\ (w^{\ast })s(t)w(t)+k(w+w^{\ast }){\mathrm{\Sigma }}_{i=2}^{\mathrm{\infty }}\frac{1}{i!}{g}^{(i)}(w^{\ast })w^i(t),\end{array}\right.$$ (2)

where

$$f(s)=\frac{s^{n_1}(t)}{s^{n_1}(t)+T_s^{n_1}},g(w)=\frac{w^{n_2}(t)}{w^{n_2}(t)+T_w^{n_2}}s(t).$$

The linearised system at $E^0$ of system (2) is given as below:

$$\left\{\begin{array}{c}\frac{\mathrm{d}p(t)}{\mathrm{d}t}=-b_{1}p(t)-b_{2}x(t)-b_{3}y(t),\\ \frac{\mathrm{d}x(t)}{\mathrm{d}t}={\beta }_{m}p(t)-b_{4}x(t)-b_{5}s(t),\\ \frac{\mathrm{d}y(t)}{\mathrm{d}t}=b_{6}x(t)+b_{5}s(t)-\alpha y(t),\\ \frac{\mathrm{d}w(t)}{\mathrm{d}t}={\beta }_{w}p(t-{\tau }_2)-{\alpha }_{w}w(t),\\ \frac{\mathrm{d}s(t)}{\mathrm{d}t}=-b_{7}w(t)-b_{8}s(t),\end{array}\right.$$ (3)

where

$$\begin{array}{rl}{b}_1& ={\alpha }_r{x}^{\ast }+{\alpha }_m{y}^{\ast },b_2={\alpha }_r{p}^{\ast },b_3={\alpha }_m{p}^{\ast },\\ b_4& =\alpha +\beta f(s^{\ast }),b_5=\beta f^{\mathrm{\prime }}(s^{\ast })x^{\ast },b_6=\beta f(s^{\ast }),\\ b_7& =k{g}^{\mathrm{\prime }}(w^{\ast })s^{\ast },b_8=kg(w^{\ast })+{\alpha }_s.\end{array}$$

The characteristic equation of system (3) can be given as below:

$$\text{det}\left(\begin{array}{ccccc}\lambda +b_1& b_2& b_3& 0& 0\\ -{\beta }_m& \lambda +b_4& 0& 0& b_5\\ 0& -b_6& \lambda +\alpha & 0& -b_5\\ -{\beta }_w{\mathrm{e}}^{-\lambda {\tau }_2}& 0& 0& \lambda +{\alpha }_w& 0\\ 0& 0& 0& b_7& \lambda +b_8\end{array}\right)=0,$$ (4)

which leads to

$$\begin{array}{c}{\lambda }^5+k_1{\lambda }^4+k_2{\lambda }^3+k_3{\lambda }^2+k_4\lambda +k_5\\ +k_6\lambda {\mathrm{e}}^{-\lambda {\tau }_2}+k_7{\mathrm{e}}^{-\lambda {\tau }_2}=0,\end{array}$$ (5)

where

$$\begin{array}{rl}{k}_1& =b_1+b_4+b_8+\alpha +{\alpha }_w,\\ k_2& =b_8\alpha +(b_8+\alpha ){\alpha }_w+b_4(b_8+\alpha +{\alpha }_w)\\ & +b_1(b_4+b_8+\alpha +{\alpha }_w)+b_2{\beta }_m,\\ k_3& =(b_1+b_4+b_8)\alpha {\alpha }_w+(b_1{b}_8+b_4{b}_8)(\alpha +{\alpha }_w)\\ & +b_1{b}_4(b_8+\alpha +{\alpha }_w)+(b_3{b}_6+b_2{b}_8+b_2\alpha +b_2{\alpha }_w){\beta }_m,\\ k_4& =b_1{b}_4{b}_8(\alpha +{\alpha }_w)+(b_1{b}_4+b_1{b}_8+b_4{b}_8)\alpha {\alpha }_w\\ & +(b_2\alpha {\alpha }_w+b_3{b}_6(b_8+{\alpha }_w)+b_2{b}_8(\alpha +{\alpha }_w)){\beta }_m,\\ k_5& =b_1{b}_4{b}_8\alpha {\alpha }_w+(b_3{b}_6+b_2\alpha )b_8{\alpha }_w{\beta }_m,\\ k_6& =(b_2-b_3)b_5{b}_7{\beta }_w,\\ k_7& =(b_3{b}_6+b_2\alpha -b_3{b}_4)b_5{b}_7{\beta }_w.\end{array}$$

We assume $i\omega$ is a root of (5) which implies that $\omega$ must satisfy the following equation:

$$\begin{array}{rl}& k_5+i{k}_4\omega -k_3{\omega }^2-i{k}_2{\omega }^3+k_1{\omega }^4+i{\omega }^5\\ & +(k_7+i{k}_6\omega )(\cos (\omega {\tau }_2)-i\sin (\omega {\tau }_2))=0.\end{array}$$ (6)

Separating up the real and imaginary parts of (6), we can get

$$\left\{\begin{array}{c}{k}_3{\omega }^2-k_1{\omega }^4-k_5=k_7\cos (\omega {\tau }_2)+k_6\omega \sin (\omega {\tau }_2),\\ k_2{\omega }^3-{\omega }^5-k_4\omega =k_6\omega \cos (\omega {\tau }_2)-k_7\sin (\omega {\tau }_2),\end{array}\right.$$ (7)

Adding up the squares of the corresponding sides of (7), we can obtain

$$\begin{array}{rl}& k_5^2-k_7^2+(k_4^2-2k_3{k}_5-k_6^2){\omega }^2+(k_3^2-2k_2{k}_4+2k_1{k}_5){\omega }^4\\ & +(k_2^2-2k_1{k}_3+2k_4){\omega }^6+(k_1^2-2k_2){\omega }^8+{\omega }^{10}=0,\end{array}$$ (8)

In addition, according to the parameters of Table 1, we can easily verify $k_4^2-2k_3{k}_5-k_6^2>0,k_3^2-2k_2{k}_4+2k_1{k}_5>0,k_2^2-2k_1{k}_3+$ $2k_4>0,k_1^2-2k_2>0$. Assuming $z={\omega }^2$, we can get

$$\begin{array}{rl}H(z)& =k_5^2-k_7^2+(k_4^2-2k_3{k}_5-k_6^2)z+(k_3^2-2k_2{k}_4+2k_1{k}_5)z^2\\ & +(k_2^2-2k_1{k}_3+2k_4)z^3+(k_1^2-2k_2)z^4+z^5=0,\end{array}$$

and

$$\begin{array}{rl}{H}^{\mathrm{\prime }}(z)& =k_4^2-2k_3{k}_5-k_6^2+2(k_3^2-2k_2{k}_4+2k_1{k}_5)z\\ & +3(k_2^2-2k_1{k}_3+2k_4)z^2+4(k_1^2-2k_2)z^3+5z^4>0.\end{array}$$

If $k_5^2<k_7^2$, (8) has at least one positive root. We use ${\omega }_{\ast }$ to represent the root of (8), and the corresponding threshold of ${\tau }_{\ast }$ is

$$\begin{array}{rl}{\tau }_{\ast }& =\frac{1}{{\omega }_{\ast }}\arccos \left(\frac{(k_3{k}_7-k_4{k}_6){\omega }_{\ast }^2+(k_2{k}_6-k_1{k}_7){\omega }_{\ast }^4}{k_7^2+k_6^2{\omega }_{\ast }^2}\right.\\ & +\left.\frac{k_6{\omega }_{\ast }^6-k_5{k}_7}{k_7^2+k_6^2{\omega }_{\ast }^2}\right).\end{array}$$

Furthermore, let $\lambda ({\tau }_2)=v({\tau }_2)+i\omega ({\tau }_2)$ be a root of (5), satisfying $v({\tau }_{\ast })=0$ and $\omega ({\tau }_{\ast })=w_{\ast }$. Next, we can prove $[\mathrm{d}\text{Re}(\lambda )/\mathrm{d}{\tau }_2]{}_{{\tau }_2={\tau }_{\ast }}>0$.

Inserting $\lambda ({\tau }_2)$ into the left‐hand side of (5) and taking derivative with respect to ${\tau }_2$, we can get

$$\begin{array}{rl}\text{Re}{\left[\frac{\mathrm{d}\lambda ({\tau }_2)}{\mathrm{d}{\tau }_2}\right]}_{{\tau }_2={\tau }_{\ast }}^{-1}& =\frac{4k_6^2{\omega }_{\ast }^{10}}{P^2+Q^2}\\ & +\frac{H^{\mathrm{\prime }}(z_{\ast })k_7^2+(k_7^2-k_5^2)k_6^2+(k_3^2-2k_2{k}_4+2k_1{k}_5)k_6^2{\omega }_{\ast }^4}{P^2+Q^2}\\ & +\frac{2(k_2^2-2k_1{k}_3+2k_4)k_6^2{\omega }_{\ast }^6+3(k_1^2-2k_2)k_6^2{\omega }_{\ast }^8}{P^2+Q^2}\end{array}$$

where

$$P=(k_4{k}_7+k_5{k}_6){\omega }_{\ast }^2-(k_2{k}_7+k_3{k}_6){\omega }_{\ast }^4+(k_1{k}_6+k_7){\omega }_{\ast }^6,$$

$$Q=k_5{k}_7-(k_3{k}_7+k_4{k}_6){\omega }_{\ast }^2+(k_1{k}_7+k_2{k}_6){\omega }_{\ast }^4-k_6{\omega }_{\ast }^6.$$

Obviously, we have

$$\text{sign}\left\{\frac{\mathrm{d}\text{Re}(\lambda )}{\mathrm{d}{\tau }_2}{|}_{{\tau }_2={\tau }_{\ast }}\right\}=\text{sign}\left\{{\frac{\mathrm{d}\text{Re}(\lambda )}{\mathrm{d}{\tau }_2}}^{-1}{|}_{{\tau }_2={\tau }_{\ast }}\right\}>0.$$

The result indicates that the system undergoes a supercritical Hopf bifurcation when ${\tau }_2={\tau }_{\ast }$ [39].

5.2 Effects of time delays when ${\tau }_1={\tau }_2=\tau$

Here, taking ${\tau }_1={\tau }_2=\tau$ as the bifurcation parameter, we can obtain the linearised characteristic equation of system (1) at the equilibrium point by Laplace transformation

$$\begin{array}{rl}& {\lambda }^5+g_1{\lambda }^4+g_2{\lambda }^3+g_3{\lambda }^2+g_4\lambda +g_5+g_6{\lambda }^3{\mathrm{e}}^{-\lambda \tau }\\ & +g_7{\lambda }^2{\mathrm{e}}^{-\lambda \tau }+g_8\lambda {\mathrm{e}}^{-\lambda \tau }+g_9{\mathrm{e}}^{-\lambda \tau }=0,\end{array}$$ (9)

where

$$\begin{array}{rl}{g}_1& =b_1+b_4+b_8+\alpha +{\alpha }_w,\\ g_2& =b_8\alpha +(b_8+\alpha ){\alpha }_w+b_4(b_8+\alpha +{\alpha }_w)\\ & +b_1(b_4+b_8+\alpha +{\alpha }_w),\\ g_3& =(b_1+b_4+b_8)\alpha {\alpha }_w+(b_1{b}_8+b_4{b}_8)(\alpha +{\alpha }_w)\\ & +b_1{b}_4(b_8+\alpha +{\alpha }_w),\\ g_4& =b_1{b}_4{b}_8(\alpha +{\alpha }_w)+(b_1{b}_4+b_1{b}_8+b_4{b}_8)\alpha {\alpha }_w,\\ g_5& =b_1{b}_4{b}_8\alpha {\alpha }_w,\\ g_6& =b_2{\beta }_m,\\ g_7& =(b_3{b}_6+b_2{b}_8+b_2\alpha +b_2{\alpha }_w){\beta }_m,\\ g_8& =(b_2\alpha {\alpha }_w+b_3{b}_6(b_8+{\alpha }_w)+b_2{b}_8(\alpha +{\alpha }_w)){\beta }_m\\ & +(b_2-b_3)b_5{b}_7{\beta }_w,\\ g_8& =(b_2\alpha {\alpha }_w+b_3{b}_6(b_8+{\alpha }_w)+b_2{b}_8(\alpha +{\alpha }_w)){\beta }_m\\ & +(b_2-b_3)b_5{b}_7{\beta }_w,\\ g_9& =b_2\alpha (b_8{\alpha }_w{\beta }_m+b_5{b}_7{\beta }_w)+b_3(b_6{b}_8{\alpha }_w{\beta }_m-b_4{b}_5{b}_7{\beta }_w\\ & +b_5{b}_6{b}_7{\beta }_w).\end{array}$$

To simplify the calculation process, we omit the repeated calculation process. By using the same method as Section 5.1, we can get the threshold ${\tau }_0$ as

$$\begin{array}{rl}{\tau }_0& =\frac{1}{{\omega }_0}\arccos \left(\frac{(g_9-g_7{\omega }_0^2)(g_5-g_3{\omega }_0^2+g_1{\omega }_0^4)}{(g_9-g_7{\omega }_0^2)(g_7{\omega }_0^2-g_9)-{(g_8{\omega }_0-g_6{\omega }_0^3)}^2}\right.\\ & +\left.\frac{(g_8{\omega }_0-g_6{\omega }_0^3)(g_4{\omega }_0-g_2{\omega }_0^3+{\omega }_0^5)}{(g_9-g_7{\omega }_0^2)(g_7{\omega }_0^2-g_9)-{(g_8{\omega }_0-g_6{\omega }_0^3)}^2}\right).\end{array}$$

Furthermore, let $\lambda (\tau )=v(\tau )+i\omega (\tau )$ be a root of (9), satisfying $v({\tau }_0)=0$ and $\omega ({\tau }_0)=w_0$, we get

$$\text{sign}\left\{\frac{\mathrm{d}\text{Re}(\lambda )}{\mathrm{d}\tau }{|}_{\tau ={\tau }_0}\right\}=\text{sign}\left\{{\frac{\mathrm{d}\text{Re}(\lambda )}{\mathrm{d}\tau }}^{-1}{|}_{\tau ={\tau }_0}\right\}>0.$$

This result demonstrates that the equilibrium $E^{\ast }$ of system (1) is stable when $0\le \tau <{\tau }_0$, conversely, the equilibrium point is unstable when $\tau >{\tau }_0$. Obviously, system (1) undergoes a supercritical Hopf bifurcation at $E^{\ast }$ when $\tau ={\tau }_0$. Therefore, the critical value ${\tau }_0$ is essential to determine the stability and oscillation of p53 network.